A famous theorem by Siu states that if is a smooth projective family of complex varieties, then the plurigenera of the fibers are independent of for all . This result plays an important role in the construction of moduli spaces for higher-dimensional varieties. In this talk I will construct smooth families of varieties , where is a DVR of positive or mixed characteristic, such that  for all sufficiently divisible. If time permits I will also present some positive results on invariance of plurigenera for certain families of good minimal models.